www.elsolucionario.net X[ m ] = N –1 ∑ x [n ]e n=0 mn – j2π N Signals and Systems with MATLAB® Applications Second Edition Steven T Karris Includes step-by-step procedures for designing analog and digital filters Orchard Publications www.orchardpublications.com www.elsolucionario.net Signals and Systems with MATLAB® Applications Second Edition Steven T Karris Students and working professionals will find Signals and Systems with MATLAB® Applications, Second Edition, to be a concise and easy-to-learn text It provides complete, clear, and detailed explanations of the principal analog and digital signal processing concepts and analog and digital filter design illustrated with numerous practical examples This text includes the following chapters and appendices: • Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation • Circuit Analysis with Laplace Transforms • State Variables and State Equations • The Impulse Response and Convolution • Fourier Series • The Fourier Transform • Discrete Time Systems and the Z Transform • The DFT and The FFT Algorithm • Analog and Digital Filters • Introduction to MATLAB • Review of Complex Numbers • Review of Matrices and Determinants Each chapter contains numerous practical applications supplemented with detailed instructions for using MATLAB to obtain quick solutions Steven T Karris is the president and founder of Orchard Publications He earned a bachelors degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a masters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has done post-master work at the latter He is a registered professional engineer in California and Florida He has over 30 years of professional engineering experience in industry In addition, he has over 25 years of teaching experience that he acquired at several educational institutions as an adjunct professor He is currently with UC Berkeley Extension Orchard Publications, Fremont, California Visit us on the Internet www.orchardpublications.com or email us: info@orchardpublications.com ISBN 0-9709511-8-3 $39.95 U.S.A www.elsolucionario.net Signals and Systems with MATLAB® Applications Second Edition Steven T Karris Orchard Publications www.orchardpublications.com www.elsolucionario.net Signals and Systems with MATLAB Applications, Second Edition Copyright © 2003 Orchard Publications All rights reserved Printed in the United States of America No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538 Product and corporate names are trademarks or registered trademarks of the Microsoft™ Corporation and The MathWorks™ Inc They are used only for identification and explanation, without intent to infringe Library of Congress Cataloging-in-Publication Data Library of Congress Control Number: 2003091595 ISBN 0-9709511-8-3 Copyright TX 5-471-562 www.elsolucionario.net Preface This text contains a comprehensive discussion on continuous and discrete time signals and systems with many MATLAB® examples It is written for junior and senior electrical engineering students, and for self-study by working professionals The prerequisites are a basic course in differential and integral calculus, and basic electric circuit theory This book can be used in a two-quarter, or one semester course This author has taught the subject material for many years at San Jose State University, San Jose, California, and was able to cover all material in 16 weeks, with 2½ lecture hours per week To get the most out of this text, it is highly recommended that Appendix A is thoroughly reviewed This appendix serves as an introduction to MATLAB, and is intended for those who are not familiar with it The Student Edition of MATLAB is an inexpensive, and yet a very powerful software package; it can be found in many college bookstores, or can be obtained directly from The MathWorks™ Inc., Apple Hill Drive , Natick, MA 01760-2098 Phone: 508 647-7000, Fax: 508 647-7001 http://www.mathworks.com e-mail: info@mathwork.com The elementary signals are reviewed in Chapter and several examples are presented The intent of this chapter is to enable the reader to express any waveform in terms of the unit step function, and subsequently the derivation of the Laplace transform of it Chapters through are devoted to Laplace transformation and circuit analysis using this transform Chapter discusses the state variable method, and Chapter the impulse response Chapters and are devoted to Fourier series and transform respectively Chapter introduces discrete-time signals and the Z transform Considerable time was spent on Chapter 10 to present the Discrete Fourier transform and FFT with the simplest possible explanations Chapter 11 contains a thorough discussion to analog and digital filters analysis and design procedures As mentioned above, Appendix A is an introduction to MATLAB Appendix B contains a review of complex numbers, and Appendix C discusses matrices New to the Second Edition This is an refined revision of the first edition The most notable changes are chapter-end summaries, and detailed solutions to all exercises The latter is in response to many students and working professionals who expressed a desire to obtain the author’s solutions for comparison with their own The author has prepared more exercises and they are available with their solutions to those instructors who adopt this text for their class The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation of presentation material www.elsolucionario.net The last major change is the improvement of the plots generated by the latest revisions of the MATLAB® Student Version, Release 13 Orchard Publications Fremont, California www.orchardpublications.com info@orchardpublications.com www.elsolucionario.net Table of Contents Chapter Elementary Signals Signals Described in Math Form 1-1 The Unit Step Function 1-2 The Unit Ramp Function 1-10 The Delta Function .1-12 Sampling Property of the Delta Function 1-12 Sifting Property of the Delta Function 1-13 Higher Order Delta Functions 1-15 Summary 1-19 Exercises 1-20 Solutions to Exercises 1-21 Chapter The Laplace Transformation Definition of the Laplace Transformation 2-1 Properties of the Laplace Transform 2-2 The Laplace Transform of Common Functions of Time .2-12 The Laplace Transform of Common Waveforms 2-23 Summary 2-29 Exercises 2-34 Solutions to Exercises 2-37 Chapter The Inverse Laplace Transformation The Inverse Laplace Transform Integral 3-1 Partial Fraction Expansion 3-1 Case where F ( s ) is Improper Rational Function ( m ≥ n ) 3-13 Alternate Method of Partial Fraction Expansion 3-15 Summary 3-18 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications i www.elsolucionario.net Exercises .3-20 Solutions to Exercises 3-22 Chapter Circuit Analysis with Laplace Transforms Circuit Transformation from Time to Complex Frequency 4-1 Complex Impedance Z ( s ) 4-8 Complex Admittance Y ( s ) 4-10 Transfer Functions 4-13 Summary .4-16 Exercises .4-18 Solutions to Exercises 4-21 Chapter State Variables and State Equations Expressing Differential Equations in State Equation Form 5-1 Solution of Single State Equations 5-7 The State Transition Matrix 5-9 Computation of the State Transition Matrix 5-11 Eigenvectors 5-18 Circuit Analysis with State Variables 5-22 Relationship between State Equations and Laplace Transform 5-28 Summary .5-35 Exercises .5-39 Solutions to Exercises 5-41 Chapter The Impulse Response and Convolution The Impulse Response in Time Domain 6-1 Even and Odd Functions of Time 6-5 Convolution 6-7 Graphical Evaluation of the Convolution Integral 6-8 Circuit Analysis with the Convolution Integral 6-18 Summary 6-20 ii Signals and Systems with MATLAB Applications, Second Edition Orchard Publications www.elsolucionario.net Exercises 6-22 Solutions to Exercises 6-24 Chapter Fourier Series Wave Analysis .7-1 Evaluation of the Coefficients .7-2 Symmetry .7-7 Waveforms in Trigonometric Form of Fourier Series 7-11 Gibbs Phenomenon 7-24 Alternate Forms of the Trigonometric Fourier Series 7-25 Circuit Analysis with Trigonometric Fourier Series 7-29 The Exponential Form of the Fourier Series 7-31 Line Spectra 7-35 Computation of RMS Values from Fourier Series 7-40 Computation of Average Power from Fourier Series 7-42 Numerical Evaluation of Fourier Coefficients 7-44 Summary 7-48 Exercises 7-51 Solutions to Exercises 7-53 Chapter The Fourier Transform Definition and Special Forms 8-1 Special Forms of the Fourier Transform 8-2 Properties and Theorems of the Fourier Transform 8-9 Fourier Transform Pairs of Common Functions 8-17 Finding the Fourier Transform from Laplace Transform 8-25 Fourier Transforms of Common Waveforms .8-27 Using MATLAB to Compute the Fourier Transform 8-33 The System Function and Applications to Circuit Analysis 8-34 Summary 8-41 Exercises 8-47 Solutions to Exercises 8-49 Signals and Systems with MATLAB Applications, Second Edition Orchard Publications iii ... at t = T and terminates at t = 2T This segment is expressed as v ( t ) = – A [ u ( t – T ) – u ( t – 2T ) ] (1.9) Line segment } has height A , starts at t = 2T and terminates at t = 3T This... for the time interval – ∞ < t < +∞ To this, it is convenient to divide the time interval into two parts, – ∞ < t < , and < t < ∞ For the time interval – ∞ < t < , the switch is open and therefore,... www.elsolucionario.net Sifting Property of the Delta Function that is, multiplication of any function f ( t ) by the delta function δ ( t ) results in sampling the function at the time instants where the delta